Properties of the ellipse 5*(x - 3)^2 = 20 - 4*(y - 6)^2

The calculator will find the properties of the ellipse $$$5 \left(x - 3\right)^{2} = 20 - 4 \left(y - 6\right)^{2}$$$, with steps shown.

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Equation:

Center:$$$($$$

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First focus:$$$($$$

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Second focus:$$$($$$

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First vertex:$$$($$$

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Second vertex:$$$($$$

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First co-vertex:$$$($$$

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Second co-vertex:$$$($$$

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Length of the major axis:

If you have the length of the semi-major axis (a), enter its value multiplied by $$$2$$$ here.

Length of the minor axis:

If you have the length of the semi-minor axis (b), enter its value multiplied by $$$2$$$ here.

Eccentricity:

Area:

First directrix:

In any form you want: $$$y = -6$$$, $$$y = 2 x + 5$$$, etc.

Second directrix:

In any form you want: $$$x - 3 = 0$$$, $$$- 8 x + 7 y + 1 = 0$$$, etc.

First point:$$$($$$

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Second point:$$$($$$

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Third point:$$$($$$

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Fourth point:$$$($$$

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Vertical major axis (parallel to the y-axis)?

Horizontal minor axis (parallel to the x-axis)?

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Your Input

Find the center, foci, vertices, co-vertices, major axis length, semi-major axis length, minor axis length, semi-minor axis length, area, circumference, latera recta, length of the latera recta (focal width), focal parameter, eccentricity, linear eccentricity (focal distance), directrices, x-intercepts, y-intercepts, domain, and range of the ellipse $$$5 \left(x - 3\right)^{2} = 20 - 4 \left(y - 6\right)^{2}$$$.

Solution

The equation of an ellipse is $$$\frac{\left(x - h\right)^{2}}{a^{2}} + \frac{\left(y - k\right)^{2}}{b^{2}} = 1$$$, where $$$\left(h, k\right)$$$ is the center, $$$b$$$ and $$$a$$$ are the lengths of the semi-major and the semi-minor axes.

Our ellipse in this form is $$$\frac{\left(x - 3\right)^{2}}{4} + \frac{\left(y - 6\right)^{2}}{5} = 1$$$.

Thus, $$$h = 3$$$, $$$k = 6$$$, $$$a = 2$$$, $$$b = \sqrt{5}$$$.

The standard form is $$$\frac{\left(x - 3\right)^{2}}{2^{2}} + \frac{\left(y - 6\right)^{2}}{\left(\sqrt{5}\right)^{2}} = 1$$$.

The vertex form is $$$\frac{\left(x - 3\right)^{2}}{4} + \frac{\left(y - 6\right)^{2}}{5} = 1$$$.

The general form is $$$5 x^{2} - 30 x + 4 y^{2} - 48 y + 169 = 0$$$.

The linear eccentricity (focal distance) is $$$c = \sqrt{b^{2} - a^{2}} = 1$$$.

The eccentricity is $$$e = \frac{c}{b} = \frac{\sqrt{5}}{5}$$$.

The first focus is $$$\left(h, k - c\right) = \left(3, 5\right)$$$.

The second focus is $$$\left(h, k + c\right) = \left(3, 7\right)$$$.

The first vertex is $$$\left(h, k - b\right) = \left(3, 6 - \sqrt{5}\right)$$$.

The second vertex is $$$\left(h, k + b\right) = \left(3, \sqrt{5} + 6\right)$$$.

The first co-vertex is $$$\left(h - a, k\right) = \left(1, 6\right)$$$.

The second co-vertex is $$$\left(h + a, k\right) = \left(5, 6\right)$$$.

The length of the major axis is $$$2 b = 2 \sqrt{5}$$$.

The length of the minor axis is $$$2 a = 4$$$.

The area is $$$\pi a b = 2 \sqrt{5} \pi$$$.

The circumference is $$$4 b E\left(\frac{\pi}{2}\middle| e^{2}\right) = 4 \sqrt{5} E\left(\frac{1}{5}\right)$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{a^{2}}{c} = 4$$$.

The latera recta are the lines parallel to the minor axis that pass through the foci.

The first latus rectum is $$$y = 5$$$.

The second latus rectum is $$$y = 7$$$.

The endpoints of the first latus rectum can be found by solving the system $$$\begin{cases} 5 x^{2} - 30 x + 4 y^{2} - 48 y + 169 = 0 \\ y = 5 \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $$$\left(3 - \frac{4 \sqrt{5}}{5}, 5\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5} + 3, 5\right)$$$.

The endpoints of the second latus rectum can be found by solving the system $$$\begin{cases} 5 x^{2} - 30 x + 4 y^{2} - 48 y + 169 = 0 \\ y = 7 \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $$$\left(3 - \frac{4 \sqrt{5}}{5}, 7\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5} + 3, 7\right)$$$.

The length of the latera recta (focal width) is $$$\frac{2 a^{2}}{b} = \frac{8 \sqrt{5}}{5}$$$.

The first directrix is $$$y = k - \frac{b^{2}}{c} = 1$$$.

The second directrix is $$$y = k + \frac{b^{2}}{c} = 11$$$.

The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).

Since there are no real solutions, there are no x-intercepts.

The y-intercepts can be found by setting $$$x = 0$$$ in the equation and solving for $$$y$$$: (for steps, see intercepts calculator).

Since there are no real solutions, there are no y-intercepts.

The domain is $$$\left[h - a, h + a\right] = \left[1, 5\right]$$$.

The range is $$$\left[k - b, k + b\right] = \left[6 - \sqrt{5}, \sqrt{5} + 6\right]$$$.

Answer

Standard form/equation: $$$\frac{\left(x - 3\right)^{2}}{2^{2}} + \frac{\left(y - 6\right)^{2}}{\left(\sqrt{5}\right)^{2}} = 1$$$A.

Vertex form/equation: $$$\frac{\left(x - 3\right)^{2}}{4} + \frac{\left(y - 6\right)^{2}}{5} = 1$$$A.

General form/equation: $$$5 x^{2} - 30 x + 4 y^{2} - 48 y + 169 = 0$$$A.

First focus-directrix form/equation: $$$\left(x - 3\right)^{2} + \left(y - 5\right)^{2} = \frac{\left(y - 1\right)^{2}}{5}$$$A.

Second focus-directrix form/equation: $$$\left(x - 3\right)^{2} + \left(y - 7\right)^{2} = \frac{\left(y - 11\right)^{2}}{5}$$$A.

Graph: see the graphing calculator.

Center: $$$\left(3, 6\right)$$$A.

First focus: $$$\left(3, 5\right)$$$A.

Second focus: $$$\left(3, 7\right)$$$A.

First vertex: $$$\left(3, 6 - \sqrt{5}\right)\approx \left(3, 3.76393202250021\right)$$$A.

Second vertex: $$$\left(3, \sqrt{5} + 6\right)\approx \left(3, 8.23606797749979\right)$$$A.

First co-vertex: $$$\left(1, 6\right)$$$A.

Second co-vertex: $$$\left(5, 6\right)$$$A.

Major axis length: $$$2 \sqrt{5}\approx 4.472135954999579$$$A.

Semi-major axis length: $$$\sqrt{5}\approx 2.23606797749979$$$A.

Minor axis length: $$$4$$$A.

Semi-minor axis length: $$$2$$$A.

Area: $$$2 \sqrt{5} \pi\approx 14.049629462081453$$$A.

Circumference: $$$4 \sqrt{5} E\left(\frac{1}{5}\right)\approx 13.318334443130703$$$A.

First latus rectum: $$$y = 5$$$A.

Second latus rectum: $$$y = 7$$$A.

Endpoints of the first latus rectum: $$$\left(3 - \frac{4 \sqrt{5}}{5}, 5\right)\approx \left(1.211145618000168, 5\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5} + 3, 5\right)\approx \left(4.788854381999832, 5\right)$$$A.

Endpoints of the second latus rectum: $$$\left(3 - \frac{4 \sqrt{5}}{5}, 7\right)\approx \left(1.211145618000168, 7\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5} + 3, 7\right)\approx \left(4.788854381999832, 7\right)$$$A.

Length of the latera recta (focal width): $$$\frac{8 \sqrt{5}}{5}\approx 3.577708763999664$$$A.

Focal parameter: $$$4$$$A.

Eccentricity: $$$\frac{\sqrt{5}}{5}\approx 0.447213595499958$$$A.

Linear eccentricity (focal distance): $$$1$$$A.

First directrix: $$$y = 1$$$A.

Second directrix: $$$y = 11$$$A.

x-intercepts: no x-intercepts.

y-intercepts: no y-intercepts.

Domain: $$$\left[1, 5\right]$$$A.

Range: $$$\left[6 - \sqrt{5}, \sqrt{5} + 6\right]\approx \left[3.76393202250021, 8.23606797749979\right]$$$A.

Properties of the ellipse $$$5 \left(x - 3\right)^{2} = 20 - 4 \left(y - 6\right)^{2}$$$

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